How would you prove mathematically that electric field inside a conductor is zero? I know the arguments but I am looking for a mathematical proof for the same.
The simplest proof is that for a conductor the current density inside the conductor is given by:
$$ \mathbf J = \sigma \mathbf E $$
where $\sigma$ is the conductivity and $\mathbf E$ is the field inside the conductor. At equilibrium the current density has to be zero everywhere inside the conductor, and this is only possible if $\mathbf E$ is zero everywhere inside the conductor.
A corollary of this is that all the charge has to reside on the surface of the conductor. If the field is zero inside the conductor then the flux through any closed Gaussian surface inside the conductor is zero, and that means the charge enclosed by that surface must be zero.
You can simply prove it by gauss law_
Take a closed surface ( inside the solid conductor) , if there is no charge within this , there will be no flux.hence no electric field.
The matter of the fact is that charge only exist on the outer surface of the conductor.( I am taking a solid conductor with no cavity)
If it would not be a conductor , then charge will somehow distributed within the material , and any closed surface has some charge within it .therefore, the field will not be zero. One always has to to assume one of the properties of conductor to prove this so I have taken the fact that charge always resides on the surface. therefore from the Gauss law there can't be any electric field inside the solid conductor